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into methyl additions, which are known for the fentanyl analogues such as α-methylfentanyl and cis-3-methylfentanyl. These analogues can possess a wide variety of modified pharmacological properties, including increased and decreased potency (receptor binding efficiency), increased or decreased half-life (metabolic bin...	{ "page_id": 46987861, "title": "List of fentanyl analogues" }
a simple procedure for organizing a potential analogue of fentanyl into the total number of unique stereoisomers, the number of true stereocenters on the molecule, and the number of Cahn-Ingold-Prelog R/S assignments that are appropriate for that analogue. The procedure used in the analysis of stereochemistry in these ...	{ "page_id": 46987861, "title": "List of fentanyl analogues" }
therefore are the self-same molecule. For this reason fentanyl does not have R/S assignments. The second case studied here is of 3-methylfentanyl. There are two potential stereocenters, at the 4-carbon and also at the 3-carbon, where there is additionally a methyl group. Now, we mark both 3 and 4 carbon as potential st...	{ "page_id": 46987861, "title": "List of fentanyl analogues" }
When we draw out all potential stereoisomers, we see that the C-4 stereocenter is super-imposable, eliminating it as a true stereocenter. This leaves only 2 R/S assignments that follow the orientation of the stereocenter at the C-α (alpha carbon) position for the real α-methylfentanyl. These are labeled 7S and 7R, a re...	{ "page_id": 46987861, "title": "List of fentanyl analogues" }
in potency between different members of the class. The weakest compounds such as benzylfentanyl are around the same potency as codeine (i.e. approximately 1/10th the potency of morphine), while the strongest compounds such as carfentanil and ohmefentanil can be over 10,000x more potent than morphine, meaning there is a...	{ "page_id": 46987861, "title": "List of fentanyl analogues" }
This page provides supplementary chemical data on barium chloride. == Material Safety Data Sheet == SIRI Science Stuff (Dihydrate) == Structure and properties == == Thermodynamic properties == == Spectral data == == References == Linstrom, Peter J.; Mallard, William G. (eds.); NIST Chemistry WebBook, NIST Standard Refe...	{ "page_id": 3013208, "title": "Barium chloride (data page)" }
Vapor-compression evaporation is the evaporation method by which a blower, compressor or jet ejector is used to compress, and thus, increase the pressure of the vapor produced. Since the pressure increase of the vapor also generates an increase in the condensation temperature, the same vapor can serve as the heating me...	{ "page_id": 5634651, "title": "Vapor-compression evaporation" }
power is between 35 and 45 kW per metric ton of compressed vapors. === Equipment for MVR evaporators === The compressor is necessarily the core of the unit. Compressors used for this application are usually of the centrifugal type, or positive displacement units such as the Roots blowers, similar to the (much smaller) ...	{ "page_id": 5634651, "title": "Vapor-compression evaporation" }
in the relevant page. The size of the other pieces of equipment, such as the main heat exchanger, the vapor head, etc. (see evaporator for details), is governed by the evaporation process. == Comparison == These two compression-type evaporators have different fields of application, although they do sometimes overlap. A...	{ "page_id": 5634651, "title": "Vapor-compression evaporation" }
psig - or more), the motive steam consumption will be in the range of 2 kg per kg of suction vapors. A higher compression ratio means a smaller heat exchanger, and a reduced investment cost. Moreover, a compressor is an expensive machine, while an ejector is much simpler and cheap. As a conclusion, MVR machines are use...	{ "page_id": 5634651, "title": "Vapor-compression evaporation" }
technologies can make clean water from sources no higher in TDS than approximately 35 g/L. For economic reasons evaporators are seldom operated on low-TDS water sources. Those applications are filled by reverse osmosis. The already brackish water which enters a typical evaporator is concentrated further. The increased ...	{ "page_id": 5634651, "title": "Vapor-compression evaporation" }
gas which in 2008 had become increasingly valuable. The water quality of evaporators is four times better which is needed for the drum boilers. The evaporators, when coupled with standard drum boilers, produce steam which is more "reliable, less costly to operate, and less water-intensive." By 2008 about 85 per cent of...	{ "page_id": 5634651, "title": "Vapor-compression evaporation" }
Genetic purging is the increased pressure of natural selection against deleterious alleles prompted by inbreeding. Purging occurs because deleterious alleles tend to be recessive, which means that they only express all their harmful effects when they are present in the two copies of the individual (i.e., in homozygosis...	{ "page_id": 40041053, "title": "Genetic purging" }
reduces the negative impact of inbreeding on fitness. If inbreeding is due just to random mating in a finite population, due to purging the fitness mean fitness declines less than would be expected just from inbreeding and, after some initial decline, it can even rebound up to almost its value before inbreeding. Anothe...	{ "page_id": 40041053, "title": "Genetic purging" }
expected to decline exponentially as inbreeding increases, where inbreeding is measured using Wright's inbreeding coefficient F (the reason why decline is exponential on F instead of linear is just that fitness is usually considered a multiplicative trait). The rate at which fitness declines as F increases (the inbreed...	{ "page_id": 40041053, "title": "Genetic purging" }
as explained below, or of the genealogy of individuals. However this requires some information on the magnitude of the deleterious effects that are hidden in the heterozygous condition but become expressed in homozygosis. The larger this magnitude, denoted purging coefficient d, the more efficient is purging. An intere...	{ "page_id": 40041053, "title": "Genetic purging" }
progressive increase of inbreeding. Then inbreeding depression occurs at a rate δ, due to (partially) recessive deleterious alleles that were present at low frequencies at different loci. This means that, in the absence of selection, the expected value for mean fitness after t generations of inbreeding, would be: W t =...	{ "page_id": 40041053, "title": "Genetic purging" }
reduces to a small value (say N=10), and remains that small for many generations. As inbreeding increases, the probability of being homozygous for one (or more) of these lethal alleles also increases, causing fitness to decline. However, as those lethals begin to occur in homozygosis, natural selection begins purging t...	{ "page_id": 40041053, "title": "Genetic purging" }
in laboratory experiments and in vertebrate populations undergoing inbreeding in zoos or in the wild, as well as in humans. The detection of purging is often obscured by many factors, but there is consistent evidence that, in agreement with the predictions explained above, slow inbreeding results in more efficient purg...	{ "page_id": 40041053, "title": "Genetic purging" }
Cathepsin zymography is a technique for quantifying enzymatic activity of the cathepsin family of cysteine proteases. It is based on SDS-PAGE whereby samples tested for cathepsin activity are loaded into a polyacrylamide gel and then separated by molecular weight. Gelatin is embedded in the gel itself, providing a subs...	{ "page_id": 36108896, "title": "Cathepsin zymography" }
== === Detection of cancer === Zymography for detection of cancer with cathepsins as biomarkers == See also == Cathepsin Zymography SDS-PAGE Cysteine protease == References ==	{ "page_id": 36108896, "title": "Cathepsin zymography" }
Promoter bashing is a technique used in molecular biology to identify how certain regions of a DNA strand, commonly promoters, affect the transcription of downstream genes. Under normal circumstances, proteins bind to the promoter and activate or repress transcription. In a promoter bashing assay, specific point mutati...	{ "page_id": 16185953, "title": "Promoter bashing" }
interactions with each other as well as the binding sites can also be assayed in this way; candidate proteins must instead be identified by protein/protein interaction assays instead of an EMSA. == Procedure == This is an example procedure for a promoter bashing assay, adapted from Boulin et al.: Clone the region of DN...	{ "page_id": 16185953, "title": "Promoter bashing" }
to reporter gene. The promoters to be assayed must be ligated to a reporter gene so that gene expression levels can be measured. The reporter gene must be a sufficient distance from the promoter that the promoter affects it as a wild-type promoter would affect a gene. This can be verified with the positive control (ful...	{ "page_id": 16185953, "title": "Promoter bashing" }
Food Weekly News is a weekly food science and agricultural newspaper reporting on the latest developments in research in food production. It is published by Vertical News, an imprint of NewsRx, LLC. == External links == Official website Articles on HighBeam Research	{ "page_id": 38730338, "title": "Food Weekly News" }
The Merck Index is an encyclopedia of chemicals, drugs and biologicals with over 10,000 monographs on single substances or groups of related compounds published online by the Royal Society of Chemistry. == History == The first edition of the Merck's Index was published in 1889 by the German chemical company Emanuel Mer...	{ "page_id": 457314, "title": "Merck Index" }
(1976) – editor Martha Windholz, a Merck chemist 10th (1983), ISBN 0-911910-27-1 – editor Martha Windholz. In 1984 the Index became available online as well as printed. 11th (1989), ISBN 0-911910-28-X 12th (1996), ISBN 0-911910-12-3 – editor Susan Budavari, a Merck chemist 13th (2001), ISBN 0-911910-13-1 – editor Marya...	{ "page_id": 457314, "title": "Merck Index" }
Emotional selection is a form of evolutionary selection where decisions are made based primarily on emotional factors. The German philosopher Ferdinand Fellmann proposed in 2009 emotional selection as a third form of evolutionary selection besides natural and sexual selection. Loving, monogamous pair-bonding seems to b...	{ "page_id": 75037284, "title": "Emotional selection (evolution)" }
In classical mechanics, the Laplace–Runge–Lenz vector (LRL vector) is a vector used chiefly to describe the shape and orientation of the orbit of one astronomical body around another, such as a binary star or a planet revolving around a star. For two bodies interacting by Newtonian gravity, the LRL vector is a constant...	{ "page_id": 719460, "title": "Laplace–Runge–Lenz vector" }
scientists discovered it. The LRL vector has been re-discovered and re-formulated several times; for example, it is equivalent to the dimensionless eccentricity vector of celestial mechanics. Various generalizations of the LRL vector have been defined, which incorporate the effects of special relativity, electromagneti...	{ "page_id": 719460, "title": "Laplace–Runge–Lenz vector" }
all central forces, but this generalized vector is a complicated function of position, and usually not expressible in closed form. The LRL vector differs from other conserved quantities in the following property. Whereas for typical conserved quantities, there is a corresponding cyclic coordinate in the three-dimension...	{ "page_id": 719460, "title": "Laplace–Runge–Lenz vector" }
Carl Runge in a popular German textbook on vectors, which was referenced by Wilhelm Lenz in his paper on the (old) quantum mechanical treatment of the hydrogen atom. In 1926, Wolfgang Pauli used the LRL vector to derive the energy levels of the hydrogen atom using the matrix mechanics formulation of quantum mechanics, ...	{ "page_id": 719460, "title": "Laplace–Runge–Lenz vector" }
m moving under the action of a fixed force. However, the same definition may be extended to two-body problems such as the Kepler problem, by taking m as the reduced mass of the two bodies and r as the vector between the two bodies. Since the assumed force is conservative, the total energy E is a constant of motion, E =...	{ "page_id": 719460, "title": "Laplace–Runge–Lenz vector" }
+ A L 2 cos ⁡ θ {\displaystyle u\equiv {\frac {1}{r}}={\frac {km}{L^{2}}}+{\frac {A}{L^{2}}}\cos \theta } where θ {\displaystyle \theta } is the angle between A and the position vector r. Further alternative formulations are given below. == Derivation of the Kepler orbits == The shape and orientation of the orbits can ...	{ "page_id": 719460, "title": "Laplace–Runge–Lenz vector" }
energy E is negative (bound orbits), the eccentricity is less than one and the orbit is an ellipse. Conversely, if the energy is positive (unbound orbits, also called "scattered orbits"), the eccentricity is greater than one and the orbit is a hyperbola. Finally, if the energy is exactly zero, the eccentricity is one a...	{ "page_id": 719460, "title": "Laplace–Runge–Lenz vector" }
= 2 m \| E \| {\textstyle p_{0}={\sqrt {2m\|E\|}}} . This circular hodograph is useful in illustrating the symmetry of the Kepler problem. == Constants of motion and superintegrability == The seven scalar quantities E, A and L (being vectors, the latter two contribute three conserved quantities each) are related by two equ...	{ "page_id": 719460, "title": "Laplace–Runge–Lenz vector" }
one-dimensional orbits in phase space, since the orbit is the intersection of the phase-space isosurfaces of their constants of motion. Consequently, the orbits are perpendicular to all gradients of all these independent isosurfaces, five in this specific problem, and hence are determined by the generalized cross produ...	{ "page_id": 719460, "title": "Laplace–Runge–Lenz vector" }
⟩ = ∂ ∂ L { 1 T ∫ 0 T h ( r ) d t } = ∂ ∂ L { m L 2 ∫ 0 2 π r 2 h ( r ) d θ } , {\displaystyle {\begin{aligned}{\frac {\partial }{\partial L}}\langle h(r)\rangle &={\frac {\partial }{\partial L}}\left\{{\frac {1}{T}}\int _{0}^{T}h(r)\,dt\right\}\\[1em]&={\frac {\partial }{\partial L}}\left\{{\frac {m}{L^{2}}}\int _{0}^...	{ "page_id": 719460, "title": "Laplace–Runge–Lenz vector" }
SO(3) × SO(3). The three components Li of the angular momentum vector L have the Poisson brackets { L i , L j } = ∑ s = 1 3 ε i j s L s , {\displaystyle \{L_{i},L_{j}\}=\sum _{s=1}^{3}\varepsilon _{ijs}L_{s},} where i=1,2,3 and εijs is the fully antisymmetric tensor, i.e., the Levi-Civita symbol; the summation index s ...	{ "page_id": 719460, "title": "Laplace–Runge–Lenz vector" }
brackets of D with the angular momentum vector L can then be written in a similar form { D i , L j } = ∑ s = 1 3 ε i j s D s . {\displaystyle \{D_{i},L_{j}\}=\sum _{s=1}^{3}\varepsilon _{ijs}D_{s}.} The Poisson brackets of D with itself depend on the sign of H, i.e., on whether the energy is negative (producing closed,...	{ "page_id": 719460, "title": "Laplace–Runge–Lenz vector" }
(where \| H \| = − H {\displaystyle \|H\|=-H} ). === Laplace-Runge-Lenz operator for the hydrogen atom in momentum space === Scaled Laplace-Runge-Lenz operator in the momentum space was found in 2022 . The formula for the operator is simpler than in position space: A ^ p = ı ( l ^ p + 1 ) p − ( p 2 + 1 ) 2 ı ∇ p , {\displa...	{ "page_id": 719460, "title": "Laplace–Runge–Lenz vector" }
the hydrogen atom == Poisson brackets provide a simple guide for quantizing most classical systems: the commutation relation of two quantum mechanical operators is specified by the Poisson bracket of the corresponding classical variables, multiplied by iħ. By carrying out this quantization and calculating the eigenvalu...	{ "page_id": 719460, "title": "Laplace–Runge–Lenz vector" }
= − m k 2 2 ℏ 2 H − 1 − I , {\displaystyle C_{1}=-{\frac {mk^{2}}{2\hbar ^{2}}}H^{-1}-I,} where H−1 is the inverse of the Hamiltonian energy operator, and I is the identity operator. Applying these ladder operators to the eigenstates \|ℓmn〉 of the total angular momentum, azimuthal angular momentum and energy operators, ...	{ "page_id": 719460, "title": "Laplace–Runge–Lenz vector" }
The peculiar symmetry of the Kepler problem results in the conservation of both the angular momentum vector L and the LRL vector A (as defined above) and, quantum mechanically, ensures that the energy levels of hydrogen do not depend on the angular momentum quantum numbers ℓ and m. The symmetry is more subtle, however,...	{ "page_id": 719460, "title": "Laplace–Runge–Lenz vector" }
of all orbitals of the same energy quantum number n. Valentine Bargmann noted subsequently that the Poisson brackets for the angular momentum vector L and the scaled LRL vector A formed the Lie algebra for SO(4). Simply put, the six quantities A and L correspond to the six conserved angular momenta in four dimensions, ...	{ "page_id": 719460, "title": "Laplace–Runge–Lenz vector" }
y, z) represent the Cartesian coordinates of the normal position vector r. The three-dimensional momentum vector p is associated with a four-dimensional vector η {\displaystyle {\boldsymbol {\eta }}} on a three-dimensional unit sphere η = p 2 − p 0 2 p 2 + p 0 2 w ^ + 2 p 0 p 2 + p 0 2 p = m k − r p 0 2 m k w ^ + r p 0...	{ "page_id": 719460, "title": "Laplace–Runge–Lenz vector" }
{\boldsymbol {\eta }}} sphere, all of which intersect the ηx axis at the two foci ηx = ±1, corresponding to the momentum hodograph foci at px = ±p0. These great circles are related by a simple rotation about the ηx-axis (Figure 8). This rotational symmetry transforms all the orbits of the same energy into one another; ...	{ "page_id": 719460, "title": "Laplace–Runge–Lenz vector" }
is not conserved, it gives rise to a conserved quantity, namely A ⋅ E {\displaystyle {\mathcal {A}}\cdot \mathbf {E} } . Further generalizing the Laplace–Runge–Lenz vector to other potentials and special relativity, the most general form can be written as A = ( ∂ ξ ∂ u ) ( p × L ) + [ ξ − u ( ∂ ξ ∂ u ) ] L 2 r ^ , {\di...	{ "page_id": 719460, "title": "Laplace–Runge–Lenz vector" }
p and r are not necessarily perpendicular. The corresponding Runge–Lenz vector is more complicated, A = 1 m r 2 ω 0 A − m r 2 E + L 2 { ( p × L ) + ( m r ω 0 A − m r E ) r ^ } , {\displaystyle {\mathcal {A}}={\frac {1}{\sqrt {mr^{2}\omega _{0}A-mr^{2}E+L^{2}}}}\left\{\left(\mathbf {p} \times \mathbf {L} \right)+\left(m...	{ "page_id": 719460, "title": "Laplace–Runge–Lenz vector" }
{d}{dt}}\left(\mathbf {p} \times \mathbf {L} \right)={\frac {d\mathbf {p} }{dt}}\times \mathbf {L} =f(r)\mathbf {\hat {r}} \times \left(\mathbf {r} \times m{\frac {d\mathbf {r} }{dt}}\right)=f(r){\frac {m}{r}}\left[\mathbf {r} \left(\mathbf {r} \cdot {\frac {d\mathbf {r} }{dt}}\right)-r^{2}{\frac {d\mathbf {r} }{dt}}\r...	{ "page_id": 719460, "title": "Laplace–Runge–Lenz vector" }
{d}{dt}}(mk\mathbf {\hat {r}} ).} Therefore, A is conserved for inverse-square central forces d d t A = d d t ( p × L ) − d d t ( m k r ^ ) = 0 . {\displaystyle {\frac {d}{dt}}\mathbf {A} ={\frac {d}{dt}}\left(\mathbf {p} \times \mathbf {L} \right)-{\frac {d}{dt}}\left(mk\mathbf {\hat {r}} \right)=\mathbf {0} .} A shor...	{ "page_id": 719460, "title": "Laplace–Runge–Lenz vector" }
r and A {\displaystyle {\mathcal {A}}} . === Hamilton–Jacobi equation in parabolic coordinates === The constancy of the LRL vector can also be derived from the Hamilton–Jacobi equation in parabolic coordinates (ξ, η), which are defined by the equations ξ = r + x , η = r − x , {\displaystyle {\begin{aligned}\xi &=r+x,\\...	{ "page_id": 719460, "title": "Laplace–Runge–Lenz vector" }
total time derivative δ L = ε d d t G ( q , t ) {\displaystyle \delta L=\varepsilon {\frac {d}{dt}}G(\mathbf {q} ,t)} corresponds to a conserved quantity Γ Γ = − G + ∑ i g i ( ∂ L ∂ q ˙ i ) . {\displaystyle \Gamma =-G+\sum _{i}g_{i}\left({\frac {\partial L}{\partial {\dot {q}}_{i}}}\right).} In particular, the conserve...	{ "page_id": 719460, "title": "Laplace–Runge–Lenz vector" }
be eliminated by instead deriving the conservation of A using an approach pioneered by Sophus Lie. Specifically, one may define a Lie transformation in which the coordinates r and the time t are scaled by different powers of a parameter λ (Figure 9), t → λ 3 t , r → λ 2 r , p → 1 λ p . {\displaystyle t\rightarrow \lamb...	{ "page_id": 719460, "title": "Laplace–Runge–Lenz vector" }
{r}} ={\frac {m}{k}}\left(\mathbf {v} \times \left(\mathbf {r} \times \mathbf {v} \right)\right)-\mathbf {\hat {r}} ,} where v is the velocity vector. This scaled vector e has the same direction as A and its magnitude equals the eccentricity of the orbit, and thus vanishes for circular orbits. Other scaled versions are...	{ "page_id": 719460, "title": "Laplace–Runge–Lenz vector" }
with different scalings and symbols. The two conserved vectors, A and B can be combined to form a conserved dyadic tensor W, W = α A ⊗ A + β B ⊗ B , {\displaystyle \mathbf {W} =\alpha \mathbf {A} \otimes \mathbf {A} +\beta \,\mathbf {B} \otimes \mathbf {B} ,} where α and β are arbitrary scaling constants and ⊗ {\displa...	{ "page_id": 719460, "title": "Laplace–Runge–Lenz vector" }
"Mysteries of the gravitational 2-body problem". Archived from the original on 2008-10-21. Retrieved 2004-12-11. Baez, John (2018). "Mysteries of the gravitational 2-body problem". Retrieved 2021-05-31. Updated version of previous source. D'Eliseo, M. M. (2007). "The first-order orbital equation". American Journal of P...	{ "page_id": 719460, "title": "Laplace–Runge–Lenz vector" }
Vowpal Wabbit (VW) is an open-source fast online interactive machine learning system library and program developed originally at Yahoo! Research, and currently at Microsoft Research. It was started and is led by John Langford. Vowpal Wabbit's interactive learning support is particularly notable including Contextual Ban...	{ "page_id": 34732646, "title": "Vowpal Wabbit" }
all data into memory The hashing trick: feature identities are converted to a weight index via a hash (uses 32-bit MurmurHash3) Exploiting multi-core CPUs: parsing of input and learning are done in separate threads. Compiled C++ code == References == == External links == Official website Vowpal Wabbit's github reposito...	{ "page_id": 34732646, "title": "Vowpal Wabbit" }
Spin is an intrinsic form of angular momentum carried by elementary particles, and thus by composite particles such as hadrons, atomic nuclei, and atoms.: 183–184 Spin is quantized, and accurate models for the interaction with spin require relativistic quantum mechanics or quantum field theory. The existence of electro...	{ "page_id": 19593829, "title": "Spin (physics)" }
Standard Model, the fundamental particles are all considered "point-like": they have their effects through the field that surrounds them. Any model for spin based on mass rotation would need to be consistent with that model. === Pauli's "classically non-describable two-valuedness" === Wolfgang Pauli, a central figure i...	{ "page_id": 19593829, "title": "Spin (physics)" }
and the system properties can be discussed in terms of "integer" or "half-integer" spin models as discussed in quantum numbers below. === In Bohmian mechanics === Spin can be understood differently depending on the interpretations of quantum mechanics. In the de Broglie–Bohm interpretation, particles have definitive tr...	{ "page_id": 19593829, "title": "Spin (physics)" }
Hamiltonian will produce an actual angular velocity, and hence an actual physical rotation – that is, a change in the phase-angle, θ, over time. However, whether this holds true for free electrons is ambiguous, since for an electron, \| S \|² is a constant ⁠ 1 / 2 ⁠ ℏ , and one might decide that since it cannot change, n...	{ "page_id": 19593829, "title": "Spin (physics)" }
any known way (in contrast to the spin direction described below). The spin angular momentum S of any physical system is quantized. The allowed values of S are S = ℏ s ( s + 1 ) = h 2 π n 2 ( n + 2 ) 2 = h 4 π n ( n + 2 ) , {\displaystyle S=\hbar \,{\sqrt {s(s+1)}}={\frac {h}{2\pi }}\,{\sqrt {{\frac {n}{2}}{\frac {(n+2...	{ "page_id": 19593829, "title": "Spin (physics)" }
The common idea that "matter takes up space" actually comes from the Pauli exclusion principle acting on these particles to prevent the fermions from being in the same quantum state. Further compaction would require electrons to occupy the same energy states, and therefore a kind of pressure (sometimes known as degener...	{ "page_id": 19593829, "title": "Spin (physics)" }
therefore the Pauli exclusion principle). Specifically, the theorem requires that particles with half-integer spins obey the Pauli exclusion principle while particles with integer spin do not. As an example, electrons have half-integer spin and are fermions that obey the Pauli exclusion principle, while photons have in...	{ "page_id": 19593829, "title": "Spin (physics)" }
electron's interaction with the surrounding quantum fields, including its own electromagnetic field and virtual particles. Composite particles also possess magnetic moments associated with their spin. In particular, the neutron possesses a non-zero magnetic moment despite being electrically neutral. This fact was an ea...	{ "page_id": 19593829, "title": "Spin (physics)" }
a particle possesses not only a magnitude (how fast the body is rotating), but also a direction (either up or down on the axis of rotation of the particle). Quantum-mechanical spin also contains information about direction, but in a more subtle form. Quantum mechanics states that the component of angular momentum for a...	{ "page_id": 19593829, "title": "Spin (physics)" }
given quantum state, one could think of a spin vector ⟨ S ⟩ {\textstyle \langle S\rangle } whose components are the expectation values of the spin components along each axis, i.e., ⟨ S ⟩ = [ ⟨ S x ⟩ , ⟨ S y ⟩ , ⟨ S z ⟩ ] {\textstyle \langle S\rangle =[\langle S_{x}\rangle ,\langle S_{y}\rangle ,\langle S_{z}\rangle ]} ...	{ "page_id": 19593829, "title": "Spin (physics)" }
result is that the spin vector undergoes precession, just like a classical gyroscope. This phenomenon is known as electron spin resonance (ESR). The equivalent behaviour of protons in atomic nuclei is used in nuclear magnetic resonance (NMR) spectroscopy and imaging. Mathematically, quantum-mechanical spin states are d...	{ "page_id": 19593829, "title": "Spin (physics)" }
, m s ⟩ = ℏ 2 s ( s + 1 ) \| s , m s ⟩ , S ^ z \| s , m s ⟩ = ℏ m s \| s , m s ⟩ . {\displaystyle {\begin{aligned}{\hat {S}}^{2}\|s,m_{s}\rangle &=\hbar ^{2}s(s+1)\|s,m_{s}\rangle ,\\{\hat {S}}_{z}\|s,m_{s}\rangle &=\hbar m_{s}\|s,m_{s}\rangle .\end{aligned}}} The spin raising and lowering operators acting on these eigenvecto...	{ "page_id": 19593829, "title": "Spin (physics)" }
(half-integer spin). The total angular momentum conserved in interaction processes is then the sum of the orbital angular momentum and the spin. === Pauli matrices === The quantum-mechanical operators associated with spin-⁠1/2⁠ observables are S ^ = ℏ 2 σ , {\displaystyle {\hat {\mathbf {S} }}={\frac {\hbar }{2}}{\bold...	{ "page_id": 19593829, "title": "Spin (physics)" }
important consequences in daily life, e.g. the periodic table of the chemical elements. === Rotations === As described above, quantum mechanics states that components of angular momentum measured along any direction can only take a number of discrete values. The most convenient quantum-mechanical description of particl...	{ "page_id": 19593829, "title": "Spin (physics)" }
_{k=-j}^{j}U_{np}^{*}U_{kq}=\delta _{pq}.} Mathematically speaking, these matrices furnish a unitary projective representation of the rotation group SO(3). Each such representation corresponds to a representation of the covering group of SO(3), which is SU(2). There is one n-dimensional irreducible representation of SU...	{ "page_id": 19593829, "title": "Spin (physics)" }
2π and α = β = 0; i.e., a full rotation about the z axis, the Wigner D-matrix elements become D m ′ m s ( 0 , 0 , 2 π ) = d m ′ m s ( 0 ) e − i m 2 π = δ m ′ m ( − 1 ) 2 m . {\displaystyle D_{m'm}^{s}(0,0,2\pi )=d_{m'm}^{s}(0)e^{-im2\pi }=\delta _{m'm}(-1)^{2m}.} Recalling that a generic spin state can be written as a ...	{ "page_id": 19593829, "title": "Spin (physics)" }
can be shown that the scalar product ⟨ ψ \| ϕ ⟩ = ψ ¯ ϕ = ψ † γ 0 ϕ {\displaystyle \langle \psi \|\phi \rangle ={\bar {\psi }}\phi =\psi ^{\dagger }\gamma _{0}\phi } is preserved. It is not, however, positive-definite, so the representation is not unitary. === Measurement of spin along the x, y, or z axes === Each of the...	{ "page_id": 19593829, "title": "Spin (physics)" }
prefer to make it real and positive.) By the postulates of quantum mechanics, an experiment designed to measure the electron spin on the x, y, or z axis can only yield an eigenvalue of the corresponding spin operator (Sx, Sy or Sz) on that axis, i.e. ⁠ħ/2⁠ or −⁠ħ/2⁠. The quantum state of a particle (with respect to spi...	{ "page_id": 19593829, "title": "Spin (physics)" }
+ u y σ y + u z σ z ) . {\displaystyle S_{u}={\frac {\hbar }{2}}(u_{x}\sigma _{x}+u_{y}\sigma _{y}+u_{z}\sigma _{z}).} The operator Su has eigenvalues of ±⁠ħ/2⁠, just like the usual spin matrices. This method of finding the operator for spin in an arbitrary direction generalizes to higher spin states, one takes the dot...	{ "page_id": 19593829, "title": "Spin (physics)" }
\psi _{y\pm }\|\psi _{z\pm }\rangle {\big \|}^{2}={\tfrac {1}{2}}.} So when physicists measure the spin of a particle along the x axis as, for example, ⁠ħ/2⁠, the particle's spin state collapses into the eigenstate \| ψ x + ⟩ {\displaystyle \|\psi _{x+}\rangle } . When we then subsequently measure the particle's spin along...	{ "page_id": 19593829, "title": "Spin (physics)" }
matrices and eigenvalues in the z-basis: Also useful in the quantum mechanics of multiparticle systems, the general Pauli group Gn is defined to consist of all n-fold tensor products of Pauli matrices. The analog formula of Euler's formula in terms of the Pauli matrices R ^ ( θ , n ^ ) = e i θ 2 n ^ ⋅ σ = I cos ⁡ θ 2 +...	{ "page_id": 19593829, "title": "Spin (physics)" }
spin states, from which four beams were observed. Thus, the nuclear spin for 23Na atoms was found to be I = 3/2. The spin of pions, a type of elementary particle, was determined by the principle of detailed balance applied to those collisions of protons that produced charged pions and deuterium. p + p → π + + d {\displ...	{ "page_id": 19593829, "title": "Spin (physics)" }
spin is as a binary information carrier in spin transistors. The original concept, proposed in 1990, is known as Datta–Das spin transistor. Electronics based on spin transistors are referred to as spintronics. The manipulation of spin in dilute magnetic semiconductor materials, such as metal-doped ZnO or TiO2 imparts a...	{ "page_id": 19593829, "title": "Spin (physics)" }
the autumn of 1925, the same thought came to Dutch physicists George Uhlenbeck and Samuel Goudsmit at Leiden University. Under the advice of Paul Ehrenfest, they published their results. The young physicists immediately regretted the publication: Hendrik Lorentz and Werner Heisenberg both pointed out problems with the ...	{ "page_id": 19593829, "title": "Spin (physics)" }
1927. The original interpretation assumed the two spots observed in the experiment were due to quantized orbital angular momentum. However, in 1927 Ronald Fraser showed that Sodium atoms are isotropic with no orbital angular momentum and suggested that the observed magnetic properties were due to electron spin. In the ...	{ "page_id": 19593829, "title": "Spin (physics)" }
The molecular formula C4H6N2 may refer to: Methylpyrazole 1-Methylpyrazole 3-Methylpyrazole Fomepizole (4-methylpyrazole) Methylimidazole 1-Methylimidazole 2-Methylimidazole 4-Methylimidazole 1,4-Dihydropyrazine	{ "page_id": 29162091, "title": "C4H6N2" }
Serratus is a large scale viroinformatics platform for uncovering the total genetic diversity of Earth's virome. Originating with the goal of uncovering novel coronaviruses that may have been incidentally sequenced by other researchers, the project expanded to encompass all RNA viruses, those which encode a viral RNA-d...	{ "page_id": 72743532, "title": "Serratus (virology)" }
Annales d'histochimie was a peer-reviewed scientific journal established in 1956. The journal covered the field of histochemistry. == External links == Record at United States National Library of Medicine	{ "page_id": 22346348, "title": "Annales d'histochimie" }
Sheath current filters are electronic components that can prevent noise signals travelling in the sheath of sheathed cables, which can cause interference. Using sheath current filters, ground loops causing mains hum and high frequency common-mode signals can be prevented. Depending on the type, sheath current filters c...	{ "page_id": 24312430, "title": "Sheath current filter" }
be used in commercial satellite receivers, since low-frequency control signals and the supply voltage for the low-noise block converter have to be transferred. === Ferrite chokes === Ferrite sheath current filters consist of a ferrite sleeve around the line or cable bundle. These are common mode chokes, damping high-fr...	{ "page_id": 24312430, "title": "Sheath current filter" }
They have no electrical isolation and cannot prevent ground loops. == See also == Sheath current == External links == How to build a homemade sheath current filter. (German)	{ "page_id": 24312430, "title": "Sheath current filter" }
Methylimidazole may refer to several related chemical compounds: 1-Methylimidazole 2-Methylimidazole 4-Methylimidazole, which is chemically distinct from, but readily interconvertable with 5-methylimidazole	{ "page_id": 29162095, "title": "Methylimidazole" }
Mechanically aided scrubbers are a form of pollution control technology. This type of technology is a part of the group of air pollution controls collectively referred to as wet scrubbers. In addition to using liquid sprays or the exhaust stream, scrubbing systems can use motors to supply energy. The motor drives a rot...	{ "page_id": 8452717, "title": "Mechanically aided scrubber" }
efficiency is low. Another mechanically aided scrubber, the induced-spray, consists of a whirling rotor submerged in a pool of liquid. The whirling rotor produces a fine droplet spray. By moving the process gas through the spray, particles and gaseous pollutants can subsequently be collected. Figure 2 shows an induced-...	{ "page_id": 8452717, "title": "Mechanically aided scrubber" }
operating costs. The performance characteristics of mechanically aided scrubbers are given in Table 1. Note: These devices are used mainly for particle collection, though can also remove gaseous pollutants from the exhaust stream. == References == == Bibliography == Bethea, R. M. 1978. Air Pollution Control Technology....	{ "page_id": 8452717, "title": "Mechanically aided scrubber" }
Mayana Zatz (Hebrew: מאיינה זץ; born July 16, 1947) is a Brazilian molecular biologist and geneticist. She is a professor at the University of São Paulo, is its Research dean. == Biography == Professor Zatz's accomplishments have been recognized and she has received many awards and prizes, including the 2000 L'Oréal-UN...	{ "page_id": 5569139, "title": "Mayana Zatz" }
that period had been abandoned. These children, who generally had a normal mental development but whose muscular problems were not treated, neither went to school nor underwent physical therapy. Therefore, in 1981, Mayana and her team founded the Brazilian Association of Muscular Dystrophy (ABIM) at the Institute of Bi...	{ "page_id": 5569139, "title": "Mayana Zatz" }
the world to find one of the genes related to a dystrophy which affects the arms and legs. They also mapped the gene responsible for the Knobloch syndrome, which causes a type of progressive blindness. == References == == External links == Curriculum Vitae. CNPq Lattes System (In Portuguese). Profile for Woman of the Y...	{ "page_id": 5569139, "title": "Mayana Zatz" }
John Robert Anderson (5 March 1928 – 26 February 2007) was an Australian chemist whose research specialised on materials science. Anderson served as Chief of the Division of Material Sciences at the Commonwealth Scientific and Industrial Research Organisation from 1970 to 1978. He attended Sydney Boys High School from ...	{ "page_id": 8845939, "title": "John Robert Anderson (chemist)" }
Hideki Sakurai (櫻井英樹, Sakurai Hideki, born May 16, 1931) is a Japanese chemist. He discovered the Sakurai reaction in 1976. == References == Mitsuo Kira (1995). "Hideki Sakurai". Journal of Organometallic Chemistry. 499 (1–2): vii–x. doi:10.1016/0022-328X(95)90915-2.	{ "page_id": 15858294, "title": "Hideki Sakurai" }
The Hanuba Hanubi Paan Thaaba (Meitei for 'Old Man and Old Woman planting Colocasia/Taro'), also known as the Hanubi Hentak! Hanuba Hentak!, is a Meitei folktale of Ancient Kangleipak (early Manipur). It is about the story of an old man, an old woman and some monkeys. == Story == Once there was a childless old couple, ...	{ "page_id": 73464440, "title": "Hanuba Hanubi Paan Thaaba" }

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